P-ADIC GEOMETRY
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P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (899–934) p-ADIC GEOMETRY P S Abstract We discuss recent developments in p-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for “compact p-adic manifolds” over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the con- struction of “universal” p-adic cohomology theories. We finish with some specula- tions on how a theory that combines all primes p, including the archimedean prime, might look like. 1 Introduction In this survey paper, we want to give an introduction to the world of ideas which the author has explored in the past few years, and indicate some possible future directions. The two general themes that dominate this work are the cohomology of algebraic vari- eties, and the local and global Langlands correspondences. These two topics are classi- cally intertwined ever since the cohomology of the moduli space of elliptic curves and more general Shimura varieties has been used for the construction of Langlands corre- spondences. Most of our work so far is over p-adic fields, where we have established analogues of the basic results of Hodge theory for “compact p-adic manifolds”, have constructed a “universal” p-adic cohomology theory, and have made progress towards establishing the local Langlands correspondence for a general p-adic reductive group by using a theory of p-adic shtukas, and we will recall these results below. However, here we wish to relay another, deeper, relation between the cohomology of algebraic varieties and the structures underlying the Langlands corresondence, a re- lation that pertains not to the cohomology of specific algebraic varieties, but to the very notion of what “the” cohomology of an algebraic variety is. Classically, the study of the latter is the paradigm of “motives” envisioned by Grothendieck; however, that vi- sion has still only been partially realized, by Voevodsky [2000], and others. Basically, Grothendieck’s idea was to find the “universal” cohomology as the universal solution to a few basic axioms; in order to see that this has the desired properties, one however needs to know the existence of “enough” algebraic cycles as encoded in the standard conjec- tures, and more generally the Hodge and Tate conjectures. However, little progress has been made on these questions. We propose to approach the subject from the other side MSC2010: primary 14G22; secondary 11S37, 11R39, 11F80, 14G20. Keywords: Perfectoid spaces, rigid-analytic geometry, p-adic Hodge theory, Shimura varieties, Langlands program, shtukas, twistors. 899 900 PETER SCHOLZE and construct an explicit cohomology theory that practically behaves like a universal cohomology theory (so that, for example, it specializes to all other known cohomology theories); whether or not it is universal in the technical sense of being the universal solution to certain axioms will then be a secondary question. This deeper relation builds on the realization of Drinfeld [1980], that in the function field case, at the heart of the Langlands correspondence lie moduli spaces of shtukas. Anderson [1986], Goss [1996], and others have since studied the notion of t-motives, which is a special kind of shtuka, and is a remarkable function field analogue of mo- tives, however without any relation to the cohomology of algebraic varieties. What we are proposing here is that, despite extreme difficulties in making sense of this, there should exist a theory of shtukas in the number field case, and that the cohomology of an algebraic variety, i.e. a motive, should be an example of such a shtuka. This picture has been essentially fully realized in the p-adic case. In the first sections of this survey, we will explain these results in the p-adic case; towards the end, we will then speculate on how the full picture over Spec Z should look like, and give some evidence that this is a reasonable picture. Acknowledgments. This survey was written in relation to the author’s lecture at the ICM 2018. Over the past years, I have benefitted tremendously from discussions with many mathematicians, including Bhargav Bhatt, Ana Caraiani, Gerd Faltings, Laurent Fargues, Ofer Gabber, Eugen Hellmann, Lars Hesselholt, Kiran Kedlaya, Mark Kisin, Arthur-César le Bras, Akhil Mathew, Matthew Morrow, Michael Rapoport, Richard Taylor, and many others. It is a great pleasure to thank all of them for sharing their insights, and more generally the whole mathematical community for being so generous, inviting, supportive, challenging, and curious. I feel very glad and honoured to be part of this adventure. 2 p-adic Hodge theory Let us fix a complete algebraically closed extension C of Qp. The analogue of a com- pact complex manifold in this setting is a proper smooth rigid-analytic variety X over C .1 Basic examples include the analytification of proper smooth algebraic varieties over C , or the generic fibres of proper smooth formal schemes over the ring of integers OC of C . More exotic examples with no direct relation to algebraic varieties are given by the Hopf surfaces X = ((A2 )rig (0; 0) )/qZ ; C n f g where q C is an element with 0 < q < 1 acting via diagonal multiplication on A2. 2 j j Recall that their complex analogues are non-Kähler, as follows from the asymmetry 0 1 1 in Hodge numbers, H (X; ΩX ) = 0 while H (X; OX ) = C . On the other hand, some other non-Kähler manifolds such as the Iwasawa manifolds do not have a p-adic analogue. 1Rigid-analytic varieties were first defined by Tate [1971], and alternative and more general foundations have been proposed by various authors, including Raynaud [1974], Berkovich [1993], Fujiwara and Kato [2006], and Huber [1996]. We have found Huber’s setup to be the most natural, and consequently we will often implicitly regard all schemes, formal schemes and rigid spaces that appear in the following as adic spaces in the sense of Huber; his category naturally contains all of these categories as full subcategories. p-ADIC GEOMETRY 901 The basic cohomological invariants of a compact complex manifold also exist in this i setting. The analogue of singular cohomology is étale cohomology H´et(X; Z`), which is defined for any prime `, including ` = p. If ` p, it follows for example from ¤ work of Huber [1996, Proposition 0.5.3], that this is a finitely generated Z`-module. For ` = p, this is also true by Scholze [2013a, Theorem 1.1], but the argument is significantly harder. i Moreover, one has de Rham cohomology groups HdR(X/C ) and Hodge cohomol- i j ogy groups H (X; ΩX ), exactly as for compact complex manifolds. These are finite- i dimensional by a theorem of Kiehl [1967]. By the definition of HdR(X/C ) as the hy- percohomology of the de Rham complex, one finds an E1-spectral sequence Eij = H j (X; Ωi ) H i+j (X/C ) 1 X ) dR called the Hodge-to-de Rham spectral sequence. In complex geometry, a basic conse- quence of Hodge theory is that this spectral sequence degenerates at E1 if X admits a Kähler metric. This assumption is not necessary in p-adic geometry: Theorem 2.1 (Scholze [2013a, Corollary 1.8], Bhatt, Morrow, and Scholze [2016, The- orem 13.12]). For any proper smooth rigid-analytic space X over C , the Hodge-to- de Rham spectral sequence Eij = H j (X; Ωi ) H i+j (X/C ) 1 X ) dR degenerates at E1. Moreover, for all i 0, i i j j i i X dimC H (X; ΩX ) = dimC HdR(X/C ) = dimQp H´et(X; Qp) : j =0 Fortunately, the Hodge-to-de Rham spectral sequence does degenerate for the Hopf surface – and the examples of nondegeneration such as the Iwasawa manifolds do not have p-adic analogues. Over the complex numbers, the analogue of the equality i i dimC HdR(X/C ) = dimQp H´et(X; Qp) follows from the comparison isomorphism between singular and de Rham cohomology. In the p-adic case, the situation is slightly more complicated, and the comparison iso- morphism only exists after extending scalars to Fontaine’s field of p-adic periods BdR. If X is only defined over C , it is nontrivial to formulate the correct statement, as there is no natural map C BdR along which one can extend scalars; the correct statement ! is Theorem 6.3 below. There is however a different way to obtain the desired equality of dimensions. This relies on the Hodge–Tate spectral sequence, a form of which is implicit in Faltings’s proof of the Hodge–Tate decomposition, Faltings [1988]. Theorem 2.2 (Scholze [2013b, Theorem 3.20], Bhatt, Morrow, and Scholze [2016, The- orem 13.12]). For any proper smooth rigid-analytic space X over C , there is a Hodge– Tate spectral sequence ij i j i+j E = H (X; Ω )( j ) H (X; Zp) Z C 2 X ) ´et ˝ p 902 PETER SCHOLZE that degenerates at E2. Here, ( j ) denotes a Tate twist, which becomes important when one wants to make everything Galois-equivariant. Note that the Hodge cohomology groups appear in the other order than in the Hodge-to-de Rham spectral sequence. Remark 2.3. If X is the base change of a proper smooth rigid space defined over a discretely valued field K C , then everything in sight carries a Galois action, and it follows from the results of Tate [1967], that there is a unique Galois-equivariant splitting of the abutment filtration, leading to a Galois-equivariant isomorphism i i i j j H (X; Qp) Q C M H (X; Ω )( j ) ; ´et ˝ p Š X j =0 answering a question of Tate [ibid., Section 4.1, Remark].